Question: All of the 5th grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$2.50$ each for students, and the group paid $$35.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$20.00$ each for teachers and $$10.50$ each for students, and the group paid $$144.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+2.5y = 35}$ ${20x+10.5y = 144}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-10y = -140}$ ${20x+10.5y = 144}$ Add the top and bottom equations together. $ 0.5y = 4 $ $ y = \dfrac{4}{0.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {5x+2.5y = 35}$ to find $x$ ${5x + 2.5}{(8)}{= 35}$ $5x+20 = 35$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 8}$ into $ {20x+10.5y = 144}$ and get the same answer for $x$ ${20x + 10.5}{(8)}{= 144}$ ${x = 3}$ There were $3$ teachers and $8$ students on the field trips.